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The design of efficient combination therapies is a difficult key challenge in the treatment of complex diseases such as cancers. The large heterogeneity of cancers and the large number of available drugs renders exhaustive in vivo or even in vitro investigation of possible treatments impractical. In recent years, sophisticated mechanistic, ordinary differential equation-based pathways models that can predict treatment responses at a molecular level have been developed. However, surprisingly little effort has been put into leveraging these models to find novel therapies. In this paper we use for the first time, to our knowledge, a large-scale state-of-the-art pan-cancer signaling pathway model to identify candidates for novel combination therapies to treat individual cancer cell lines from various tissues (e.g., minimizing proliferation while keeping dosage low to avoid adverse side effects) and populations of heterogeneous cancer cell lines (e.g., minimizing the maximum or average proliferation across the cell lines while keeping dosage low). We also show how our method can be used to optimize the drug combinations used in sequential treatment plans—that is, optimized sequences of potentially different drug combinations—providing additional benefits. In order to solve the treatment optimization problems, we combine the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm with a significantly more scalable sampling scheme for truncated Gaussian distributions, based on a Hamiltonian Monte-Carlo method. These optimization techniques are independent of the signaling pathway model, and can thus be adapted to find treatment candidates for other complex diseases than cancers as well, as long as a suitable predictive model is available. 

Tree-form sequential decision making (TFSDM) extends classical one-shot decision making by modeling tree-form interactions between an agent and a potentially adversarial environment. It captures the online decision-making problems that each player faces in an extensive-form game, as well as Markov decision processes and partially-observable Markov decision processes where the agent conditions on observed history. Over the past decade, there has been considerable effort into designing online optimization methods for TFSDM. Virtually all of that work has been in the full-feedback setting, where the agent has access to counterfactuals, that is, information on what would have happened had the agent chosen a different action at any decision node. Little is known about the bandit setting, where that assumption is reversed (no counterfactual information is available), despite this latter setting being well understood for almost 20 years in one-shot decision making. In this paper, we give the first algorithm for the bandit linear optimization problem for TFSDM that offers both (i) linear-time iterations (in the size of the decision tree) and (ii) O(T−−√) cumulative regret in expectation compared to any fixed strategy, at all times T. This is made possible by new results that we derive, which may have independent uses as well: 1) geometry of the dilated entropy regularizer, 2) autocorrelation matrix of the natural sampling scheme for sequence-form strategies, 3) construction of an unbiased estimator for linear losses for sequence-form strategies, and 4) a refined regret analysis for mirror descent when using the dilated entropy regularizer. 

A two-part tariff is a pricing scheme that consists of an up-front lump sum fee and a per unit fee. Various products in the real world are sold via a menu, or list, of two-part tariffs---for example gym memberships, cell phone data plans, etc. We study learning high-revenue menus of two-part tariffs from buyer valuation data, in the setting where the mechanism designer has access to samples from the distribution over buyers' values rather than an explicit description thereof. Our algorithms have clear direct uses, and provide the missing piece for the recent generalization theory of two-part tariffs. We present a polynomial time algorithm for optimizing one two-part tariff. We also present an algorithm for optimizing a length-L menu of two-part tariffs with run time exponential in L but polynomial in all other problem parameters. We then generalize the problem to multiple markets. We prove how many samples suffice to guarantee that a two-part tariff scheme that is feasible on the samples is also feasible on a new problem instance with high probability. We then show that computing revenue-maximizing feasible prices is hard even for buyers with additive valuations. Then, for buyers with identical valuation distributions, we present a condition that is sufficient for the two-part tariff scheme from the unsegmented setting to be optimal for the market-segmented setting. Finally, we prove a generalization result that states how many samples suffice so that we can compute the unsegmented solution on the samples and still be guaranteed that we get a near-optimal solution for the market-segmented setting with high probability.